Approximation of weak adjoints by reverse automatic differentiation of BDF methods (original) (raw)
Related papers
2020
Inverse problems are of utmost importance in many fields of science and engineering. In the variational approach inverse problems are formulated as PDE-constrained optimization problems, where the optimal estimate of the uncertain parameters is the minimizer of a certain cost functional subject to the constraints posed by the model equations. The numerical solution of such optimization problems requires the computation of derivatives of the model output with respect to model parameters. The first order derivatives of a cost functional (defined on the model output) with respect to a large number of model parameters can be calculated efficiently through first order adjoint sensitivity analysis. Second order adjoint models give second derivative information in the form of matrix-vector products between the Hessian of the cost functional and user defined vectors. Traditionally, the construction of second order derivatives for large scale models has been considered too costly. Consequent...
International Journal for Numerical Methods in Engineering, 1979
The application of finite element methods to parabolic partial differential equations leads to large linear systems of first-order ordinary differential equations. Very often these systems are stiff and difficulties arise in their numerical solution. We attempt to analyse the problem of how to select numerical methods for the solution of such linear systems.
A Family of Eulerian-Lagrangian Localized Adjoint Methods for Multi-Dimensional Advection-Reaction E
1998
We develop a family of Eulerian-Lagrangian localized adjoint methods for the solution of the initial-boundary value problems for rst-order advection-reaction equations on general multi-dimensional domains. Di erent tracking algorithms, including the Euler and Runge-Kutta algorithms, are used. The derived schemes naturally incorporate in ow boundary conditions into their formulations and do not need any articial out ow boundary condition. They are fully mass conservative. Moreover, they have regularly structured, well-conditioned, symmetric and positive-de nite coe cient matrices, which can be solved e ciently by, for example, the conjugate gradient method in an optimal order number of iterations without any preconditioning needed. Numerical results are presented to compare the performance of these methods with many well studied and widely used methods, including the upwind nite di erence method, the Galerkin and the Petrov-Galerkin nite element methods with backward-Euler or Crank-Nicolson temporal discretization, and the streamline di usion nite element methods.
Eulerian-Lagrangian localized adjoint method: The theoretical framework
Numerical Methods for Partial Differential Equations, 1993
This is the second of a sequence of papers devoted to applying the localized adjoint method (LAM), in space-time, to problems of advective-diffusive transport. We refer to the resulting methodology as the Eulerian-Lagrangian localized adjoint method (ELLAM). The ELLAM approach yields a general formulation that subsumes many specific methods based on combined Lagrangian and Eulerian approaches, so-called characteristic methods (CM). In the first paper of this series the emphasis was placed in the numerical implementation and a careful treatment of implementation of boundary conditions was presented for one-dimensional problems. The final ELLAM approximation was shown to possess the conservation of mass property, unlike typical characteristic methods. The emphasis of the present paper is on the theoretical aspects of the method. The theory, based on Herrera's algebraic theory of boundary value problems, is presented for advection-diffusion equations in both one-dimensional and multidimensional systems. This provides a generalized ELLAM formulation. The generality of the method is also demonstrated by a treatment of systems of equations as well as a derivation of mixed methods.
Adjoint computations by algorithmic differentiation of a parallel solver for time-dependent PDEs
Journal of Computational Science, 2020
A computational fluid dynamics code is differentiated using algorithmic differentiation (AD) in both tangent and adjoint modes. The two novelties of the present approach are 1) the adjoint code is obtained by letting the AD tool Tapenade invert the complete layer of message passing interface (MPI) communications, and 2) the adjoint code integrates time-dependent, non-linear and dissipative (hence physically irreversible) PDEs with an explicit time integration loop running for ca. 10 6 time steps. The approach relies on using the Adjoinable MPI library to reverse the non-blocking communication patterns in the original code, and by controlling the memory overhead induced by the time-stepping loop with binomial checkpointing. A description of the necessary code modifications is provided along with the validation of the computed derivatives and a performance comparison of the tangent and adjoint codes.
International Journal for Numerical Methods in Engineering, 2009
We describe a novel variational formulation of inverse elasticity problems given interior data. The class of problems considered is rather general and includes, as special cases, plane deformations, compressibility and incompressiblity in isotropic materials, 3D deformations, and anisotropy. The strong form of this problem is governed by equations of pure advective transport. The variational formulation is based on a generalization of the adjoint-weighted variational equation (AWE) formulation, originally developed for flow of a passive scalar. We describe how to apply AWE to various cases, and prove several properties. We prove that the Galerkin discretization of the AWE formulation leads to a stable, convergent numerical method, and prove optimal rates of convergence. The numerical examples demonstrate optimal convergence of the method with mesh refinement for multiple unknown material parameters, graceful performance in the presence of noise, and robust behavior of the method when the target solution is
The purpose of this study is to show some mathematical aspects of the adjoint method that is a numerical method for the Cauchy problem, an inverse boundary value problem. The adjoint method is an iterative method based on the variational formulation, and the steepest descent method minimizes an objective functional derived from our original problem. The conventional adjoint method is time-consuming in numerical computations because of the Armijo criterion, which is used to numerically determine the step size of the steepest descent method. It is important to find explicit conditions for the convergence and the optimal step size. Some theoretical results about the convergence for the numerical method are obtained. Through numerical experiments, it is concluded that our theories are effective.
Adjoint sensitivity in PDE constrained least squares problems as a multiphysics problem
Compel-the International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2012
Purpose-The purpose of this paper is to provide a framework for the implementation of an adjoint sensitivity formulation for least-squares partial differential equations constrained optimization problems exploiting a multiphysics finite elements package. The estimation of the diffusion coefficient in a Poisson-type diffusion equation is used as an example. Design/methodology/approach-The authors derive the adjoint formulation in a continuous setting allowing to attribute to the direct and adjoint states the role of different fields to be solved for. They are one-way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. Having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post-processing procedure. This sensitivity can then be used to efficiently solve the least-squares problem. Findings-The authors derived the adjoint formulation in a continuous setting allowing the direct and adjoint states to be attributed the role of different fields to be solved. They are one-way coupled through the mismatch between measured and direct states acting as a source term in the adjoint equation. It is found that, having solved for the direct and adjoint state, the sensitivity of the cost function with respect to the design variables can then be obtained by a suitable post-processing procedure. Research limitations/implications-This paper implies that modern multiphysics finite elements packages provide a flexible and extendable software environment for the experimentation with different adjoint formulations. Such tools are therefore expected to become increasingly important in solving notoriously difficult partial differential equation (PDE)-constrained least-squares problems. The framework also provides the possibility of experimentation with different regularization techniques (total variation and multiscale techniques for instance) to handle the ill-posedness of the problem. Originality/value-In this paper the adjoint sensitivity computation is casted as a multiphysics problem allowing for a flexible and extendable implementation.